for all \(0\le x\le \varepsilon \) , where \(S_n=\sum _{i=1}^n \sigma _i\)  is total spin. It means that \(|S_n|\)  is
close to \(bn/\beta \)  with high probability. On the other hand,
all points on the hypersphere of radius \(bn/\beta \) 
will have equal probability due to symmetry.
Based on these facts, they considered
the fluctuations of the squared-length of total spin:
 
Wn:= n(2n2 b2 |Sn |2 -1 ),

where \(S_n=\sum _{j=1}^n \sigma _j\) . Let
 
B2=42(1-f'(b))b2[1-(N-1)f(b)b-(f(b))2 ].

Kirkpatrick and Nawaz [1]} proved
that when \(\beta >N\) ,
the bounded-Lipschitz distance between \(W_n/B\)  and \(Z\)  is bounded by \(C (\log n/n)^{1/4}\) .
Their proof is based on Stein's method for exchangeable pairs
(see Stein [2]}).
Recall that a random vector \((W,W^{\prime })\)  is called an exchangeable pair if \((W,W^{^{\prime }})\)  and \((W^{^{\prime }}, W)\)  have the same distribution.
Kirkpatrick and Nawaz [1]} construct an exchangeable pair as follows.
Let \(W_n\)  be as in () and let \(\sigma ^{\prime } = \lbrace \sigma _1^{\prime },\ldots ,\sigma _n^{\prime }\rbrace \) ,
where for each \(i\)  fixed, \(\sigma _i^{\prime }\)  is an independent copy of \(\sigma _i\)  given \(\lbrace \sigma _j, j\ne i\rbrace \) , i.e., given \(\lbrace \sigma _j, j\ne i\rbrace \) ,
\(\sigma _i^{\prime }\)  and \(\sigma _i\)  have the same distribution and \(\sigma _i^{\prime }\)  is conditionally independent of \(\sigma _i\) 
(see, e.g., [4]}).
Let \(I\)  be a random index independent of all others and uniformly distributed over \(\lbrace 1,\ldots ,n\rbrace \) , and let
 
Wn' = n(2n2 b2|Sn'|2 - 1 ),

where \(S_n^{\prime } = \sum _{j=1}^n\sigma _j-\sigma _I + \sigma _I^{\prime }\) .
Then \((W_n,W_{n}^{^{\prime }})\)  is an exchangeable pair (see Kirkpatrick and Nawaz [1]}, Kirkpatrick and Meckes [6]}).
